Suppose ψ1 and ψ2 are two eigenfunctions of a particle and ε1 and ε2 are the corresponding eigenvalues. If the state is in the superposition Ψ = αψ1 + βψ2 at time t=0, it evolves in time by the equation Ψ = αψ1ei ħ/ε1 t + βψ2ei ħ/ε2 t. I am trying to understand the probability amplitude Ψ*Ψ.If you take Ψ* and multiply with Ψ in the cross terms of ψ1 and ψ2 there will be the exponential part containing the imaginary component i. How is this possible? Shouldnt the probability function, being the square of the wave function always be real so that we have measurable quantity associated with the wave function? If the probability function stops being real, what does that mean in the real world? Thanks!!!